#185439

order	512
nilpotency class	3
isoclinism	isoclinic class of #185279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,6)(3,4)(7,8)(9,10,14,19,15,17,13,18)(11,20,22,12,16,24,21,23);(1,2,5,7)(3,6,4,8)(9,11,15,16)(10,12,17,23)(13,22,14,21)(18,24,19,20);(3,4)(6,8)(9,12)(10,16)(11,17)(13,20)(14,24)(15,23)(18,22)(19,21);(1,3,5,4)(2,6,7,8)(10,17)(12,23)(18,19)(20,24)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92635);